Something else that I have been thinking about is how did Harry Potter ended up with the used and very informative potions book that was owned by The Half-Blood Prince.
As we discovered in Harry Potter and the Half Blood Prince that:
It was Professor Severus Snape that had called himself that back in his school years at Hogwarts.
Also, the name wasn’t widely known as in Harry Potter and the Half Blood Prince, Harry asked Remus Lupin if he had heard of the Half Blood Prince?
Remus then told Harry that he hadn’t.
We also know that Albus Dumbledore asked Professor Snape to help Draco Malfoy and perhaps Professor Severus Snape was also trying to help Harry as well.
I also, don’t think that Severus Snape would leave a school textbook of his in a storeroom.
So, I am thinking that perhaps Professor Snape had put his old textbook there if he knew that his old textbook was going to be given to Harry to use.
As maybe perhaps to make sure that Harry was the one that gets his textbook and not Ron, he might have used the Imperious Curse on Horace Slughorn.
As perhaps Professor Snape would’ve wanted Harry to know that he had called himself The Half Blood Prince, and he did get the chance to tell Harry that he was The Half Blood Prince: near the end of Harry Potter and the Half Blood Prince.
Also no one knew who The Half Blood Prince was although Hermione did some research and suggested that the person might’ve been a female.
She also found out that there was an Eileen Prince who was Severus Snape’s Mother.
However, there were people who knew that a dark spell called Sectumsampra was a spell that Severus Snape used, and he asked Harry to give him the book. Then perhaps why he asked for the book as he might’ve expected Harry had been using his old textbook all year and Professor Snape might not have wanted it traced back to him.
Though Harry hid The Half Blood Prince’s textbook in a cupboard that was in The Room of Requirement, and it was highly likely that this book was destroyed by Fiend Fire which had happened later on in Harry Potter and the Deathly Hallows.